Phase field method to optimize dielectric devices for electromagnetic wave propagation

Transcription

1 Phase field method to optimize dielectric devices for electromagnetic wave propagation Akihiro Takezawa 1,, Mitsuru Kitamura 1 a Division of Mechanical Systems and Applied Mechanics, Institute of Engineering, Hiroshima University, Kagamiyama, Higashi-Hiroshima, Hiroshima, Japan Abstract We discuss a phase field method for shape optimization in the context of electromagnetic wave propagation. The proposed method has the same functional capabilities as the level set method for shape optimization. The first advantage of the method is the simplicity of computation, since extra operations such as re-initialization of functions are not required. The second is compatibility with the topology optimization method due to the similar domain representation and the sensitivity analysis. Structural shapes are represented by the phase field function defined in the design domain, and this function is optimized by solving a time-dependent reaction diffusion equation. The artificial double-well potential function used in the equation is derived from sensitivity analysis. We study four types of 2D or 2.5D (axisymmetric) optimization problems. Two are the classical problems of photonic crystal design based on the Bloch theory and photonic crystal wave guide design, and two are the recent topics of designing dielectric left-handed Corresponding author. Tel: ; Fax: addresses: akihiro@hiroshima-u.ac.jp (Akihiro Takezawa), kitamura@naoe.hiroshima-u.ac.jp (Mitsuru Kitamura) Preprint submitted to Journal of Computational Physics June 4, 2014

2 metamaterials and dielectric ring resonators. Keywords: Phase field, Shape optimization, Electromagnetic system, Sensitivity analysis, Topology optimization, Level set 1. Introduction Geometrical optimization has been studied to solve the problem of what shape is optimal in various problems. Recent development of optimization enables us to optimize geometry drastically, including topology. In recent studies, these methods have been used not only to optimize the existing structure but also generate a new shape having a specified function. These uses of optimization have contributed to the development of pure and applied physics (e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]). The contribution of geometrical optimization seems to be strongest for electromagnetic wave propagation, since research on finding geometry having new electromagnetic characteristics is active, as reviewed in [11]. Topology optimization is the most popular geometrical optimization method [12, 13, 14]. The basic idea is replacing the shape optimization problem by a two-phase material distribution problem consisting of an original material and an ersatz material mimicking voids. With this formulation this method can generate holes for the target domain. However, since the original problem is ill-posed, homogenization is applied and the original problem is represented as a composite material optimization problem, namely, an optimization problem of a volume fraction of these materials. This representation raises the further problem of how to obtain clear shapes from the optimal density distribution [13, 14]. 2

3 Recently, the level set method for geometrical optimization [15, 16, 17, 18] has been proposed to avoid the drawbacks described above. In this method, the target configuration is represented as a zero contour of the level set function, and the front moves along its normal direction according to the Hamilton-Jacobi equation. The moving direction of the front is decided by using shape sensitivity, which is used in classical shape optimization [19, 20] as the velocity field for the Hamilton-Jacobi equation. Level set methods allow topological changes (limited to eliminating holes), significantly improving structural performance. Moreover, this method is free from re-meshing, since the level set function is defined in an Eulerian coordinate system. The level set method was originally proposed by Osher and Sethian [21, 22, 23] as a numerical method for tracking free boundaries according to the mean curvature motion, and the mathematical background was subsequently clarified by several researchers [24, 25, 26]. The level set method has been applied to many optimization problems (e.g. [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]), including electromagnetic wave propagation. As with the level set method, the phase field method is capable of handling the motion caused by domain states such as temperature and the motion caused by the domain shape, such as the mean curvature motion, and so can also be applied to geometrical optimization. The phase field method was developed as a way to represent the surface dynamics of phase transition phenomena such as solid-liquid transitions. Research on such physical modeling can be traced back to Cahn and Hilliard [38] and Allen and Cahn [39]. The mathematical fundamentals for these physical models were constructed by several researchers [40, 41, 42]. In the early stage of this research, contribu- 3

4 tions to computing actual phase transition phenomena were made by [43, 44]. Outlines of the above history and methodologies can be found in several comprehensive reviews [45, 46, 47]. Bourdin and Chambolle [48] were the first to propose applying the phase field method to structural optimization, when it had initially been used to implement perimeter constraints [49, 50]. After that, several useful structural optimization methods were developed that incorporate the phase field method (e.g. [51, 52, 53, 54]). Another shape optimization technique has been proposed using the phase field method for simulation with a moving boundary [55, 56]. (Hereafter, when phase field method is used in the sense of shape optimization, it mentions this methodology.) This method has the almost same function as in the level set method, meaning it is a shape optimization based on a moving boundary accepting the topology change. This method has two advantages. The first is that it does not need to re-initialize the function as needs to be done in the conventional level-set method. The second is that it is very compatible with topology optimization. The domain in the phase field method is represented by a density function, the same as in the topology optimization. The front motion is decided by the sensitivity with respect to the variable defined on the domain and not the boundary. Thus, despite the fact that the phase field method is basically a shape optimization method different from topology optimization, the sensitivity analysis framework developed by topology optimization can be applied to the phase field method. A disadvantage of the method is the thickness of the front domain corresponding to the actual boundary of the object. It must have a thickness including several grids to represent smooth motion of the boundary. This is not a serious 4

5 problem in a 2D problem but cannot be overlooked in 3D problems. Since above methods were mainly developed through linear elastic static and dynamic problems, another physical problem should be solved to confirm the utility of the methods. We choose electromagnetic wave propagation as the second target physical problem. First, the 2D method is a useful way to analyze a polarized in-plane wave device, especially in the field of photonics. Thus, the drawback of the phase field method with respect to 3D analysis is not serious. Second, optimization of the electromagnetic wave device is mainly done by topology optimization as mentioned at the start of the introduction. The phase field method can easily follow this analysis due to its compatibility with topology optimization. Moreover, few studies have been reported on using the phase field method to solve the electromagnetic problem. In this research, we study the phase field method for shape optimization of electromagnetic wave propagation. First, we review the phase field method for shape optimization proposed in [55]. The structural shape is represented by the phase field function defined in the design domain containing the optimal configuration. The numerical computation is performed over the whole domain as the distribution optimization problem of two phase materials, as in conventional topology optimization. We optimize the phase field function by using a time-dependent reaction diffusion equation called the Allen-Cahn equation. An artificial double-well potential used in the equation is derived from sensitivity analysis to move the front in the direction decreasing the objective function. Then Maxwell s equations are used as the state equations. We explain the four types of 2D or 2.5D (axisymmetric) optimization prob- 5

6 lems in this paper. Two are classical topics, photonic crystal design based on the Bloch theory and photonic crystal wave guide design, and two are recent topics, design of dielectric left-handed metamaterials and dielectric ring resonators. We perform sensitivity analysis for each problem to construct the optimization procedure. The numerical results are finally shown for each problem to show the validity of the proposed method. 2. Phase field method for shape optimization 2.1. Phase field method We define a phase field function ϕ(x) over an entire analysis domain to represent the phase of the local points therein, as shown in Fig.1. From a physical point of view, the phase field function provides the average phase of the local points. Consider a closed system composed of two phases, one of which corresponds to the value α of the phase field function while the other corresponds to the value β (α < β). We represent the boundary of each phase as a smooth function that interpolates the different values ϕ, which we term the diffuse interface. The Van der Waals free energy of the system is given by F (ϕ) = Ω ( ) ε 2 2 ϕ 2 + P (ϕ) dx, (1) where ε > 0 is a coefficient determining the effect of each term. The first term represents the interaction energy term of the field in mean field theory, and the second term represents a double well potential with the value P (α) = P (β) = 0. The double well potential indicates that there are lower free energy values with minima corresponding to each phase. 6

7 ξ Ω Phase B φ β P h a s e B Phase A E d g e o f d i f f u se i n t er f ac e r eg i o n C en t er o f d i f f u se i n t er f ac e r eg i o n P h a s e A α o ξ D i f f u s e i n t e r f a c e x (a) (b) Figure 1: Examples of the phase field function. (a) A 2D domain represented by the phase field function. (b) A 1D illustration of the phase field function. Next, we consider the time-dependent evolution equation of the phase field function ϕ. The change of the phase field function with respect to time is assumed to be linearly dependent upon the direction in which the free energy function is minimized: ϕ t = K(ϕ)δF (ϕ) δϕ, (2) where K(ϕ) is a variation rate function. Substituting Eq.(1) into Eq.(2), we obtain ϕ t = K(ϕ) ( ε 2 2 ϕ P (ϕ) ). (3) Eq.(3) is known as the Allen-Cahn equation [39]. According to this equation, the front moves in its normal direction at a speed determined by the difference between each minimum of the double well potential P (ϕ) and the curvature of the diffuse interface as follows [57, 58, 59]: v = s + 1 t H + O ( 1 t 2 ), t >> 1, (4) 7

8 where s is the speed due to the difference between each minimum of the double well potential P (ϕ), and H is the mean curvature of the diffuse interface. If the potential has equal minima, the motion is only governed by the mean curvature Domain representation of the target domain in the phase field method for shape optimization We use a general shape optimization problem to explain the phase field method. Let Ω be the domain that varies during the optimization process, with the state of the domain Ω represented by some partial differential equations. The boundary Ω of Ω is divided into two boundaries, a boundary Ω D with Dirichlet boundary conditions and a boundary Ω N with Neumann boundary conditions. The state variable u is calculated based on the state equations that have these boundary conditions. We introduce the extended design domain D that contains Ω. Here, a set of admissible shapes in D can be represented as U ad = {Ω D Ω R d }. (5) Thus, the shape optimization problem for Ω is defined as minimize Ω U ad J(Ω), (6) where J(Ω) is a functional with respect to state variable u whose value depends on the shape of Ω. We represent the shapes of optimized domains using a phase field function as shown in Fig.2. The phase field function ϕ(x)(0 ϕ 1) is defined in the domain D. We consider a setting where the domain Ω 1 (x D ϕ(x) = 1) corresponds to the optimized shape Ω and the domain Ω 0 (x D ϕ(x) = 8

9 0) corresponds to D \ Ω. However, this setting is insufficient because a diffuse interface region exists when the phase field method is used. Let ξ represent the diffuse interface region. The domain representation of D is then formulated as ϕ = 1 x Ω 1, 0 < ϕ < 1 x ξ, (7) ϕ = 0 x Ω 0, where (Ω 1 ξ) Ω and (Ω 0 ξ) D \ Ω. (8) That is, the original domain Ω is represented as a subset of the union of Ω 1 and ξ. In the above setting, the position of the boundary Ω is unclear except that it lies in ξ. However, as explained in Section 2, the diffuse interface region becomes very thin when ε is very small, in which case ξ can be regarded as approximately representing Ω. We can easily pick out a clear shape in a plot by choosing an arbitrary contour value such as ϕ = 0.5 in the diffuse interface. To achieve optimization of Ω in the phase field method, we consider the distribution of a physical property of the target domain as the design variable and clip the shape of the domain from the physical property distribution. Let us consider a domain with a phase field function ϕ defined over it, and the state u is dominated by a PDE having a term with coefficient A(ϕ) that is a function of ϕ. The three states Ω 1, Ω 0 and ξ are considered in this domain according to Eq.(7). Ω 1 is filled with a material whose physical property is A, and Ω 0 is assumed to be filled with a material having a very small one that mimics a void to avoid singularities in the calculation. In addition, 9

10 the material in the diffuse interface ξ has a virtual physical property using an interpolation function E(ϕ) defined in the range E min E(ϕ) E max. Accordingly, we represent the distribution of the physical property E of the entire domain as E max, if x Ω 1. E (ϕ) = E(ϕ) (E min < E(ϕ) < E max ), if x ξ. (9) E min, if x Ω 0. D D \ Ω D Ω 0 (φ = 0) Ω Ω 1 (φ = 1 ) Ω ξ(0<φ <1 ) (a) An original domain (b) The domain represented by the phase field function Figure 2: The domain representation by the phase field function 2.3. Evolution of the phase field function The phase field function evolves with a virtual time t in the interval t 1 t t 2, corresponding to a descent step of the function in the optimization problem. The evolution equation is formulated as ϕ t = κ 2 ϕ P (ϕ), (t 1 t t 2 ), (10) ϕ n = 0 on D, 10

11 where κ = ε 2 is a positive coefficient of the diffusion term and P (ϕ) is a double well potential. As explained in Section 2, when the phase field function follows Eq.(10), the diffuse interface moves in a normal direction with the velocity determined by the difference between the minima of the double well potential. The double well potential P (ϕ) is determined by sensitivity analysis to achieve motion in a direction that reduces the value of the objective function. The conditions that the double well potential P (ϕ) must satisfy are as follows: P (0) = 0, P (1) = aj (ϕ t1 ), P (0) = P (1) = 0 (a > 0), (11) where J (ϕ) is the sensitivity of the objective function with respect to ϕ, ϕ t1 is the value of ϕ at time t 1 and a is its positive coefficient. A sketch of the double well potential is shown in Fig.3. Since the function evolves in the direction of the smaller minimum of the double well potential, the objective function can be minimized in the same way as for conventional steepest-descent methods. Moreover, since the sensitivity distribution depends on the interpolation function E(ϕ) in Eq.(9), the boundary motion of the proposed methodology depends on the interpolation function the same way as with topology optimization. 11

12 P ( φ ) 1 aj ( φ ) ' t 1 φ Figure 3: Double well potential Note that some methods have been proposed that apply the perimeter minimization of the phase field method to perimeter control of topology optimization methods [48, 51]. We remark that the primary difference between these methods and the proposed methodology is in their origin and functions. Since the aforementioned methods came from topology optimization, which updates the density function based on sensitivity analysis, nucleation of holes in the target structure is possible. Conversely, the proposed methodology aims for the same outcome using the level set method for shape optimization, whose roots are in classical shape optimization based on boundary variation. Thus, there are no hole nucleation mechanisms in our method. Because of this characteristic, the proposed methodology has an initial dependency unlike the topology optimization. 3. Phase field method for dielectric domain optimization in electromagnetic wave propagation 3.1. Domain representation In the phase field method, the shape of the dielectric domain is represented as the distribution of the phase field function. The optimization is 12

13 performed by representing the physical property as a function of the phase field function ϕ. The relative permittivity over the design domain ϵ is expressed as the function of the phase field function ϕ, (0 ϕ 1) using the permittivity of the original dielectric material ϵ 1 and that of air ϵ 0 according to Eq.(9). The optimal shape of the dielectric domain can then be specified as a distribution in ϕ. Both ϵ 0 and ϵ 1 are assumed to be isotropic. The interpolation function used in the phase field interface is decided according to the target problem Equations of state The 3D vector Helmholtz equations are the equations of state representing resonance-mode wave propagation within the dielectric domain with relative permittivity ϵ and the same permeability as that of open space. Assuming a time harmonic solution with angular frequency ω, we write the Helmholtz equation for the electric field E derived from Maxwell s equation as ( E) + ϵ ω2 E = 0, (12) c2 where c is the speed of light. Conversely, the Helmholtz equation for the magnetic field H is written as ( ) 1 ϵ H + ω2 H = 0. (13) c2 H and E can be calculated from the following Maxwell-Ampère equation without the current density term: H = ϵ E t. (14) The following boundary conditions are introduced for the domain governed by the above equations depending on the target problem. We show 13

14 only the formulations with respect to the electric field are shown here. By replacing E by H, we can applied them to the magnetic field. Floquet periodic boundary condition: E = E src e jk r per on Γ pe, (15) First-order absorbing boundary condition: n ( E) + ikn (n E) = 0 on Γ ab, (16) First-order absorbing boundary with incident wave: n ( E) + ikn (n E) = 2ikn (n E inc ), (17) Perfect electric (magnetic) wall: E n = 0 on Γ pw. (18) where E src is the electric field vector on the source boundary in the periodic boundary condition, k is the wave number vector with respect to the coordinate system, r per is a vector perpendicular to the periodic boundaries with a magnitude equal to the distance between them, n is the unit vector normal to the boundary, k is the wave number in free-space and E inc is the incident electric field vector Optimization problems We now apply the optimization method to designing photonic crystal unit cells, photonic crystal wave guides, dielectric left-handed metamaterials (LHMs) and dielectric ring resonators Design of photonic crystal Our first case study is designing the unit cell of a 2D photonic crystal [60], which is the classical problem in designing dielectric devices for electromagnetic wave propagation first discussed in [61]. In this problem, we consider 14

15 periodic structures that extend along the z direction as shown in Fig.4. The shape of a unit cell on the xy plane is designed to prevent light propagation from all directions in a certain frequency range. We limit the target wave to a transverse-magnetic (TM) wave with an electric field only in the z direction and a magnetic field only in the xy plane (E = [0, 0, E z ] T, H = [H x, H y, 0] T ). We assume that the square unit cell has the mirror symmetrical shape on the horizontal center line, vertical center line and diagonal center line. Due to the symmetry, the wave vectors k are considered on only the irreducible Brillouin zone [60] shown in Fig.4. The design domain is also set to this zone. That is, the distribution of the dielectric material is optimized there, and the resulting shape is repeated in other areas of the unit cell. Periodic dielectric material Design target Μ Γ Χ y z y x x Figure 4: An outline of a 2D photonic crystal composed of dielectric columns homogeneous along the z direction and its unit cell. The triangle in the unit cell represents the Brillouin zone. The photonic band gap is calculated as the gap between the upper and the lower eigenfrequencies of the target order in every possible k in the irre- 15

16 ducible Brillouin zone. The calculation is performed by solving the Helmholtz equation in Eq.(12) only for E z as the eigenvalue problem under the Floquet periodic boundary condition in Eq.(15) on 2D domain. We evaluate the performance of the photonic crystal using the absolute difference between the two dimensionless frequencies formulated as (ω target+1 ω target ) d/2πc [1, 4], where ω target and ω target+1 represent the lower and upper angular eigenfrequencies of the optimized gap and d is the length of the unit cell. (On the other hand, the band gap is evaluated as a relative value by dividing the gap by the sum of the maximum value of the lower frequency and the minimum value of the upper frequency in [5]).) Thus, we define the objective function for photonic crystal design as maximizing the gap between the eigenvalues Λ = ω 2. The optimization is formulated as the following unconstrained problem: minimize J 1 (ϕ) = [min{λ target+1 (ϕ, k)} max{λ target (ϕ, k)}]. (19) This objective function contains the effects minimizing the maximum value of the target order eigenvalue and maximizing the minimum value of the target+1 order eigenvalue independently of the existence of the gap. Thus, if there is no gap in the target structure, the optimization decreases the band crossing represented as a positive objective function. After the objective function becomes zero, which means the band crossing is zero, the objective function becomes negative and the band is generated. Note that the described objective function works for cases where the upper band lies above the lower band for all k-values, that is, where the target and target+1 order eigenvalues can be distinguished. If not, numerical techniques are required to specify these eigenvalues. 16

17 Design of photonic crystal wave guide Second, we study design optimization of the photonic crystal wave guide. The design domain is set to the corner of the L-shaped photonic crystal wave guide as shown in Fig.5. This problem was also studied in the early stage of electromagnetic wave propagation optimization [2]. In this problem, we optimize the distribution of the dielectric material for the target wave, which is limited to a TM wave as in the first example. The calculation is performed by solving the Helmholtz equation in Eq.(12) only for E z as a frequency response problem for the input wave in a 2D domain. For designing photonic crystal wave guides, we use the integrated time-averaged Poynting vector in the specified domain for the objective function. By summing the Poynting vector corresponding to several frequencies in the optimization, the optimization problem is formulated as the unconstrained problem minimize J 2 (ϕ) = = i Jˆ 2i (ϕ) m i m D. (20) C 1 (x)re(e i H i )dx, where m represents the number of frequencies considered in the optimization, E i and H i are the electric and magnetic fields corresponding to the i-th input frequencies, C 1 (x) is a coefficient function denoting the location of the target domain and the component of the Poynting vector, Re( ) and Im( ) represent respectively the real and imaginary parts of the variable and the bar expresses the conjugate. 17

18 Periodic dielectric material Output wave Design target Input wave y x Figure 5: An outline of a 2D photonic crystal waveguide composed of dielectric columns homogeneous along the z direction Design of dielectric left-handed metamaterials As a case study dealing with a recent topic in optimizing electromagnetic wave propagation, we consider the design of a dielectric 2D LHM with negative permeability studied in [62]. Dielectric LHMs have an advantage over conventional ones using metallic split-ring resonators or thin wires in terms of avoiding metallic loss and wider effective frequency regions. The LH behavior can be observed in periodic dielectric media as shown in Fig.4. A Mie resonance mode occurring at a certain frequency provides the LH characteristic for periodic media (e.g. [63, 64, 65]). In this problem, the wave is a transverse-electric (TE) wave with a magnetic field only in the z direction and electric field only in the xy plane (E = [E z, E y, 0] T, H = [0, 0, H z ] T ). We analyze the device with the unit cell model considering periodicity. The 18

19 Helmholtz equation for the magnetic field H simplified to a TE wave is solved on the domain as a frequency response problem for a wave incident from the left. The right side is considered as an output side. We set a firstorder absorbing boundary condition to both the input and output sides. A perfect electric wall is set to the upper and the lower sides to represent the periodicity. To evaluate the performance of the LHM, we use the following effective permeability [66] formulated by the S-parameters: µ eff = n eff Z eff, (21) where n eff = 1 { } 1 kd cos 1 (1 S S 2 21 ), (22) S 21 (1 + S 11 ) Z eff = 2 2 S 21 (1 S 11 ) 2 2 S, (23) 21 and k is the wave number, d is the length of a unit cell, n eff is the effective refractive index and Z eff is the effective impedance. S 11 and S 21 are S-parameters formulated as Γ S 11 = in (E E1 inc ) E1 inc ds, (24) Γ in E1 inc 2 ds Γ S 21 = out E E2 sc ds Γ out E2 sc 2 ds, (25) where Γ in and Γ out are the input and output boundaries of the wave corresponding to ports 1 and 2 respectively, and E inc 1 and E sc 2 are incident and scattering electric fields on the input and output boundaries respectively. The objective function is formulated to make the real or imaginary part of 19

20 the effective permeability by minimizing each as follows: minimize J 3 (ϕ) = Re(µ eff ) or Im(µ eff ). (26) Design of dielectric ring resonator The above three design problems are performed in a 2D domain. As a 2.5D (axisymmetric) design problem, we perform the cross-sectional design of dielectric ring resonators in a cylindrical coordinate system studied in [67]. The optical modes, set up in these devices by emissions from input light, form circular continuous closed beams governed by internal reflections along the boundary of the resonator as shown in Fig.6(a). The mode shape with the circular electric field distribution is called a whispering-gallery (WG) mode. This type of resonator has potential applications to low-threshold microcavity lasers and light-matter systems for quantum networking [68, 69, 70]. Optical resonators exploiting WG modes have attained high performance levels such as high quality factor (Q factor), which measures the inverse of the energy decay rate, and low mode volume, which signifies the spatial confinement of the light [69]. 20

21 z Design target Emission wave φ Input wave o r (a) (b) Figure 6: (a) An outline of a dielectric ring resonator. (b) The analysis and design domain in cylindrical coordinates. In this problem, TM waves are considered as emissions. We use cylindrical coordinates to model a WG ring resonator centered on the origin in free space as illustrated in Fig.6(b). The coordinate system consists of the axial, azimuthal, and radial coordinates. We treat the vertical cross-sectional device shape in the z-r plane as the design target and obtain the final axisymmetric form by generating the solid of revolution. The Helmholtz equation for the magnetic field H is solved as an eigenvalue problem in the cylindrical coordinate system. To suppress spurious modes in the analysis, we introduce a weak penalty term [71] to yield the following modified equation used in [72]: ( ) 1 ϵ H p ( H) + ω2 H = 0, (27) c2 where p is a coefficient. The first term describes perfect magnetic conduction 21

22 and the latter term first-order absorption on the boundary. The state variable is the set comprising the time-dependent radial, azimuthal, and axial components of the magnetic field vector H(r, t). We factorize the azimuthal dependence from the variable to obtain H(r) = e imϕ [H r (r, z), ih ϕ (r, z), H z (r, z)] T, (28) where M is the azimuthal mode order. The performance criteria in designing a WG ring resonator are the Q factor and mode volume. In calculating the Q factor, we consider only the radiation loss, for which Q rad is calculated as follows [69, 73]: Q rad = Re(f) 2Im(f). (29) In the analysis for the domain surrounded by the perfect matched layer (PML) domain, the real part of the frequency represents the total energy of the domain, whereas the imaginary part represents the rate of energy absorption by the PML domain, i.e. the radiation loss (Chapter 5 in [74]). In addition, the mode volume is formulated as [69, 73] D V mode = ϵ E 2 dx max(ϵ E 2 ). (30) During optimization, we target the following three tasks: 1. Maximizing the Q factor associated with emissions as expressed by Eq.(29). 2. Minimizing the mode volume as expressed by Eq.(30). 3. Specifying the emission wavelength λ = c/re(f). 22

23 We first fulfill task 3 by introducing the corresponding equality constraint. Assuming normalized eigenmodes H ( D H 2 dx = 1) in the eigenfrequency analysis, the total electric energy over the analysis domain equals the square of the angular eigenfrequency ( D ϵ E 2 dx = Re(ω) 2 ) [75, 76]. That is, by pre-specifying the emission wavelength and azimuthal mode order, both numerators in the expressions for the Q-factor and mode volume are constant during optimization. Thus, the objective functions for Task 1, J Q, and Task 2, J V, and the equality constraint h for Task 3 are formulated as or minimize J Q (ϕ) = Im(f), (31) ϕ subject to minimize J V (ϕ) = max(ϵ E 2 ), (32) ϕ with λ 0 denoting the specified wavelength. h(ϕ) = λ λ 0 = 0, (33) 3.4. Sensitivity analysis The gap of the double well potential P (ϕ) is set based on sensitivity analysis of the objective function with respect to ϕ, which is the function defined on D. Here, we show only the results of the sensitivity analysis and explain the detailed derivation in the appendix. The sensitivity of the i-th eigenvalue Λ i used in the objective function in Eq.(19) is given by Λ i(ϕ) = ϵ (ϕ) Λ i c 2 E i E i, (34) 23

24 where E i is the i-th eigenvector. As explained in the appendix, the sensitivity of the eigenvalue can be calculated without adjoint variables. Note that, when the eigenvalue is repeated, it is not differentiable in the normal sense. In this case, only directional derivatives can be obtained [77, 78]. The sensitivity of the Poynting vector in the specified domain in Eq.(20) is calculated as Ĵ 2 (ϕ) = ϵ (ϕ) ω2 E p, (35) c2 where p is the adjoint variable calculated from the following adjoint equations: { C 1 1 D 2 Re v + D ( 1 ( v) ( p) dx ) jωµ E D + E ϵ(ϕ) ω2 c 2 v pdx ( 1 ) } jωµ v dx Γ n {v ( p)} ds = 0 (36) n p + ikn n p = 0 on Γ ab and Γ inc. (37), The sensitivity of the S-parameters used in calculating the effective permeability in Eq.(21) is calculated with ( S 11(ϕ) 1 ϵ = (ϕ) ( H) Γ in E1 inc 2 ds ϵ 2 ( ϵ S 12(ϕ) (ϕ) = ( p) H ϵ 2 ) H, (38) ), (39) 24

25 where p is the adjoint variable calculated from the adjoint equations 1 v E Γ out E2 sc 2 2 sc ds ds Γ out, + ( v) ( p) dx ϵ(ϕ) ω2 D D c v pdx n {v ( p)} ds = 0, 2 Γ (40) n p + ikn n p = 0 on Γ in and Γ out, (41) ( 1 ) jωµ p n = 0 on Γ pw. (42) In the case of the ring resonator design problem, the sensitivity of the eigenfrequency is required in Eq.(31), which is calculated from the sensitivity of the i-th eigenvalue Λ i given by ( ) ϵ Λ (ϕ) i(ϕ) = ( H i ) H i. (43) The maximum local energy density used in the objective function in Eq.(32) is calculated by using a function C 2 (x) having the value 1 at the maximum energy point and the value 0 at other points: ϵ 2 max(ϵ E 2 ) ϕ = C 2(x)ϵ E i 2 ϕ = C. (44) 2(x)µ H i 2. ϕ Thus, its sensitivity is obtained as max (ϵ E i 2 ) ϕ ( ) ϵ (ϕ) = ( p) H D ϵ 2 i, (45) 25

26 where p is the adjoint variable calculated from the following adjoint equations ( ) 1 C 2 (x)v pdx ( v) D D ϵ p dx, Λ i c v pdx p( p)( v)dx n {v ( p)} ds = 0 2 D D Γ (46) n p + ikn n p = 0 on Γ ab. (47) 4. Numerical implementation 4.1. Algorithm Based on the above formulation, the optimization algorithm is constructed as follows. 1. Set the initial value of the phase field function ϕ expressing the initial shape of domain Ω. 2. Iterate the following procedure until convergence. (a) Calculate the state variable u and adjoint state p with respect to ϕ n at the n-th iteration by solving the state equation and the adjoint equation using the finite element method. (b) Calculate the objective function and constraints. (c) Calculate the sensitivity of the objective function and prepare the evolution equation for ϕ. (d) Calculate the updated value of ϕ by solving the evolution equation. 3. Obtain the optimal shape of Ω expressed as the optimal distribution of ϕ. 26

27 4.2. Setting the evolution equation We set the double well potential satisfying Eq.(11). We set it to be P (ϕ) = W (x)w(ϕ) + G(x)g(ϕ), (48) where w(ϕ) = ϕ 2 (1 ϕ 2 ), g(ϕ) = ϕ 3 (6ϕ 2 15ϕ + 10), (49) which are the same as those used in [79]. w(ϕ) is a function satisfying w(0) = w(1) = w (0) = w (1) = 0, and g(ϕ) is one with g(0) = 0, g(1) = 1 and g (0) = g (1) = 0. Sketches of these functions are shown in Fig.7. W (x) and G(x) are coefficients of these functions. W (x) determines the height of the wall of the double well potential, which affects the thickness of the diffuse interface. W (x) is set to be 1 4 here. The value of G(x) is chosen to be G(x) = aj (ϕ t1 ) as explained section 2.3. To avoid complicated parameter settings due to the order difference of J (ϕ) depending on the optimization problem, we first normalize the sensitivity by dividing by its L 2 -norm, and the new coefficient η is set as Substituting Eqs.(48)-(50) into Eq.(10), we obtain G(x) = η J (ϕ t1 ) J (ϕ t1 ). (50) ϕ t = κ 2 ϕ ϕ ( 1 = κ 2 ϕ = κ 2 ϕ + ϕ(1 ϕ) (t 1 t t 2 ). (W (x)w(ϕ) + G(x)g(ϕ)) 4 w (ϕ) + η J ) (ϕ t1 ) J (ϕ t1 ) g (ϕ) { ϕ η J (ϕ t1 ) (1 ϕ)ϕ J (ϕ t1 ) }, (51) 27

28 w g 1 o 1 φ o 1 φ (a) (b) Figure 7: Sketches of functions (a) w and (b) g Numerical method for the evolution equation The finite difference method is used to solve Eq.(51) numerically. discretize the time derivative term using the forward Euler scheme and solve the equation with the explicit scheme. The time step t is restricted by the following Courant-Friedrichs-Lewy (CFL) condition for stable convergence in the 2D case: We ( t κ ( x) + t ) 1 2 ( y) 2 2, (52) where t > 0 is the time step and x and y are space steps in the x and y directions respectively. Aside from the above CFL condition, we must consider the convergence of the reaction term. Here, the reaction term is discretized by a so-called semi-implicit scheme [79], in which forward time terms are partly included. In the 2D case, let ϕ n i,j be the value of ϕ at the n-th iteration at the point x i,j. The scheme then leads to the following discretization: 28

29 ϕ n+1 i,j ϕ n i,j t ( ϕ n i 1,j 2ϕ n i,j + ϕ n i+1,j = κ + ϕn i,j 1 2ϕ n i,j + ϕ n ) i,j+1 ( x) 2 ( y) 2 ϕ n+1 i,j (1 ϕ n i,j)r(ϕ n i,j) for r(ϕ n i,j) 0 + ϕ n i,j(1 ϕ n+1 i,j )r(ϕ n i,j) for r(ϕ n i,j) > 0, (53) where r(ϕ n i,j) = ϕ n i,j η J (ϕ t1 ) J (ϕ t1 ) ϕn i,j(1 ϕ n i,j). (54) This discretization guarantees that ϕ remains in the interval 0 ϕ 1 even when the time step is large. Although the forward time term ϕ n+1 is included in the right side of the above equation, ϕ n+1 can obviously be calculated easily without solving a linear system, and the computational cost is almost equal to that for the ordinary explicit scheme Handling of equality constraint Normally, we use the sensitivity in Eq.(50) to choose the double well potential gap that decides the moving direction of the phase field interface. This way, the constraint is embedded in the objective function using the Lagrange multiplier method [18]. However, when handling equality constraints like Eq.(33), the solution of Eq.(51) oscillates around the limit of the equality constraint and does not converge easily to the optimal solution. Here, as a special numerical technique for handling the equality constraint, we decide the potential gap based on sequential linear programing (SLP). SLP is one of the simplest methods that can handle the equality constraint problem directory. Let us consider the minimization of the objective function J(X) with 29

30 an equality constraint h(x) = 0 with respect to a design variable vector X = [X 1, X 2,..., X n ]. In SLP, the increment X of the design variable X is obtained as the solution of the following linear programming problem: minimize J(X) + J(X) T X, (55) subject to h(x) + h(x) T X = 0, (56) X low X i X up for i = 1,..., n, (57) where X low and X up are the lower and upper bounds of X i and the upper suffix T denotes the transpose. It is only in designing dielectric ring resonators that the above linear programming problem is solved in each optimization iteration using the discretized phase field function as the design variable, and the obtained increment is used to form the double well potential gap. Note that in this method the obtained increment of the phase field function works only on the phase field interface. Thus, unlike the ordinary SLP method, the equality constraint might not always be satisfied, although this method worked well in the numerical example of this study. 5. Numerical examples To confirm the validity of the proposed method, we study the four numerical examples described in Section 3. In all examples, we regard a material representing a void as air with relative permittivity ϵ 0 = 1. We ignore the frequency dependence of the relative permittivity of the dielectric material. 30

31 The relative permeabilities of the both air and the dielectric material are set to 1. The coefficient of normalized sensitivity η in Eq.(50) is set to 20. At each iteration, we perform a finite element analysis of the state equation and one update of the evolution equation for the phase field function by solving the finite difference equation of the semi-implicit scheme. The time step t is set to be 0.9 times the limit of the Courant number. All optimal configurations are plotted as the distribution of the phase field function of the optimal results. All finite element analyses are performed with a commercial software, COMSOL Multiphysics, using the second-order Lagrange element. The design domain is discretized by the square element that is used for both finite element and finite difference analyses. We set the evaluation point of the phase field function on the center of the square element. Let the grid interval of the finite difference mesh be l. The diffusion coefficient κ in Eq.(51) is based on the value 0.1 l Design of photonic crystal We perform the first optimization for designing the unit cell of a photonic crystal. One unit cell is modeled with the finite element method. The length d of the unit cell is set to 1 as a dimensionless parameter. Since we assume mirror symmetries in the unit cell, we only regard the area shown in Fig.8 as the design domain, and resulting configurations are copied to other areas on the unit cell. In the design domain, the relative permittivity is formulated as a function of the phase field: { ( 1 1 ϵ(ϕ) = + ϕ 1 )} 1. (58) ϵ 0 ϵ 1 ϵ 0 31

32 The relative permittivity ϵ 1 of the dielectric material is set to 8.9 (alumina). The unit cell domain is meshed with square elements. The diffusion coefficient κ is set to 0.5 l 2. We calculate the wave vector k of the Floquet periodic boundary condition in Eq.(15) in the triangular irreducible Brillouin zone, with a step size of We maximize the band gap above the first-, second- and third-order eigenfrequencies. The initial shape and the corresponding dispersion diagram are shown in Fig.8. The initial shape does not have any band gaps in the depicted frequency range. Periodic boundary condition Μ Γ Χ Frequency ωd/2πc Γ Χ Μ Γ (a) (b) Figure 8: (a) Initial shape for the photonic crystal design problem. The inset triangle shows the irreducible Brillouin zone and the design domain. (b) The dispersion diagram of the initial shape. Because the first and second eigenvalues and the first, second and third eigenvalues are repeated at points X and M respectively in the initial shape, we calculated all possible directional derivatives and accepted the best one for updating the density function at the first iteration to identify target eigen- 32

33 values. Figure 9 shows the convergence histories of the objective functions until convergence with 50 iterations. The variation of the objective function from negative to positive indicates generation of the band gap through the first few optimization iterations. Figure 10(a), (c) and (e) shows the 3 3 array of the optimal configuration of the unit cell of the last iteration for the three cases. Figure 10(b), (d) and (f) shows the calculated dispersion diagram for the shape. Each dispersion diagram shows the dimensionless frequency ωd/2πc. The band gaps were certainly obtained above the specified eigenfrequency, although the optimization started with no band gap shape. The optimal configuration for the first band gap is nearly 0.2 circle radius, which is the reference shape in [60]. The optimal configurations of the second and third band gaps were also similar to the one proposed by [4], although the size of each part is different because of the difference in permittivity of the dielectric material. These comparisons also confirm the validity of the proposed method. The increases in the objective function after extracting black and white shapes are 0.84%, 0.60% and 1.83%. These values show the modeling error caused by the shape representation of the field function. Fortunately, our optimization did not encounter any eigenvalue crossing problem [80] around the target band gap order in any examples studied. However, it could occur depending on the initial shape. One way to overcome this is introducing a so-called mode tracking technique (e.g. [80]). 33

34 0.3 Objective function First band Second band Third band Iteration Figure 9: Convergence history of the objective function of the photonic crystal optimization. 34

35 Frequency ωd/2πc Band gap = Γ Χ Μ Γ (a) (b) Frequency ωd/2πc Band gap = Γ Χ Μ Γ (c) (d) 0.7 Frequency ωd/2πc Band gap = Γ Χ Μ Γ (e) 35 (f) Figure 10: Optimal configurations and dispersion diagrams. (a) and (b) First band. (c) and (d) Second band. (e) and (f) Third band.

36 5.2. Design of photonic crystal wave guide For the second optimization we design the shape of the corner part of a photonic crystal wave guide. The analysis domain is shown in Fig.11. The dielectric material is located periodically over the domain, and the wave guide is constructed by removing it according to the intended wave path. The length d of the unit cell is 0.57µm. The incident wave enters from the port on the left bottom side, and the output port is set on the upper right side. The design domain is set on the corner of the wave guide. We use the upper right part of the domain as the reference domain for evaluating the Poynting vector. In the design domain, we formulate the relative permittivity the same way as in the previous example using Eq.(58). We set the relative permittivity ϵ 1 of the dielectric material to 11.4 (GaAs). The design domain is meshed with square elements. Other domains are meshed with triangular elements. The diffusion coefficient κ is set to 0.2 l 2. Since the preferred range of the dimensionless frequencies ωd/2πc is from 0.35 to 0.42 for this wave guide [60], those of the incident waves are set to 0.38, 0.39 and 0.40 as a middle range. We set the electric field of the incident wave to the constant value 1 on the boundary. The initial shapes are set to two types as shown in Fig.12 considering the initial dependency of the proposed method. The first is a periodic dielectric material, which does not propagate the wave. The second is the reference shape reported in [60], which directs wave propagation in the specified range. 36

37 1.14µm 0.57µm First order absorbing boundary Reference domain 0.57µm 0.855µm 0.855µm 0.57µm 5.13µm 0.57µm 1.14µm First order absorbing boundary with incident wave Design domain 2.28µm 2.28µm 0.285µm 0.285µm 5.13µm Figure 11: Analysis domain for designing the photonic crystal wave guide including the design domain. 37

38 (a) (b) Figure 12: Two initial shapes for designing the photonic crystal wave guide. (a) Periodic dielectric materials. (b) Reference shape. Figure 13 shows convergence histories of objective functions until 50 iterations. Figure 14 shows optimal configurations after 41 iterations from the initial shape shown in Fig.12(a) and 44 iterations from the initial shape shown in Fig.12(b). The objective functions calculated from Fig.14 (a) and (b) are W and W (2.8% improvement from the reference shape), which are similar results. Although the optimization started from the shape that does not direct wave propagation, the optimal shapes that propagate the wave with the specified frequency were obtained in Fig.14 (a). Figure 15 shows the electric field distribution for the incident wave with the dimensionless frequency 0.39 in both optimal configurations. We observed a smooth bending of the electric field in both cases. Figure 16 shows the integrated Poynting vector in the reference domain when the dimensionless frequency of the incident wave is from 0.35 to Both shapes achieve slightly higher values than the reference shape in the middle frequency range 38

39 from 0.38 to The changes in the objective function after extracting black and white shapes are 0.30% and 0.60%. 1x From Fig.12 (a) From Fig.12 (b) Objective function -1x x x x Iteration Figure 13: Convergence history of the objective function of the photonic crystal wave guide optimization. (a) (b) Figure 14: Optimal configurations for the photonic crystal wave guide. (a) From initial shapes shown in Fig.12 (a). (b) From initial shapes shown in Fig.12 (b). 39

40 (a) (b) Figure 15: Electric field distribution with the incident wave with dimensionless frequency (a) Optimal configuration shown in Fig.14 (a). (b) Optimal configuration shown in Fig.14 (b). Time-averaged Poynting vector 4x x10 2x10-16 Figure 12(b) -16 1x10 Figure 14(a) Figure 14(b) Dimensionless frequency ωd/2πc Figure 16: Integrated Poynting vectors in the reference domain of the optimal and reference configurations 5.3. Design of dielectric left-handed metamaterial For the third optimization we design a dielectric LHM. The design target is assumed to be a periodic arrangement of dielectric poles as in the first 40

41 example. The analysis domain is set as a square unit cell including the inner square design domain shown in Fig.17. The length d of the unit cell is 120µm. Assuming mirror symmetry along the x and y axes, the optimization is performed on the upper right side of the design domain. We set the length of the design domain to 80µm. To represent the periodicity along the y direction, we set a perfect electric wall at the top and bottom sides of the domain. Both sides are given a first-order absorbing boundary with a wave incident on the left side and scattered to the right side along the x direction. We formulate the relative permittivity in the design domain as ϵ(ϕ) = ϵ 0 + ϕ (ϵ 1 ϵ 0 ). (59) The relative permittivity ϵ 1 of the dielectric material is set to 100 i as a virtual material including a little loss for the numerical stability as done in [62]. The analysis domain is meshed with square elements. The electric field of the incident wave is set to the constant value 1 on the boundary. We set the target frequency, which we use as the frequency of the incident wave, to 0.3 THz. The initial shape is set to a 60µm diameter circle. The diffusion coefficient κ is set to 0.1 l 2. We first minimize the imaginary part of the effective permeability. Then, we optimize the real part. 41

42 Perfect electric wall First order absorbing boundary with incident wave 120µm Design domain 80µm First order absorbing boundary y 120µm x Figure 17: Analysis model for designing the dielectric LHM including the design domain. Figure 18 shows convergence histories of the real and imaginary part of the effective permeability. We performed 94 and 16 iterations for the imaginary and real part optimizations respectively. Figure 19 shows the optimal configurations for the imaginary and real parts of the effective permeability. These results are similar to those obtained in [62]. Figure 20 shows the magnetic field distributions and electric field directions in these optimal configurations. Figure 21 shows the effective permeability for both configurations obtained by varying the frequency of the input wave from 0.2 THz to 0.5 THz. The imaginary and real parts of the effective permeability take the negative peaks at the specified frequency. So-called Mie resonance modes are observed in Fig.20 (c) and (d), which has a localized distribution of the magnetic field on the center of the dielectric material. This is the main mode for the dielectric LHM. We see in Fig.20 (a) and (b) that although the hot spot of the magnetic field is slightly off the center of the dielectric material after we optimize the imaginary part, the hot spot is exactly in the center after we optimize the real part. The change in the objective function after 42

43 extracting black and white shapes is 1.85%. 20 Imaginary part optimization Real part optimization Effective permeability Re(µ eff ) Im(µ eff ) Iteration Figure 18: Convergence history of the effective permeability of the dielectric metamaterial optimization. (a) (b) Figure 19: Optimal configurations obtained after optimizing (a) the imaginary part and (b) the real part of the effective permittivity. 43

44 (a) (b) (c) (d) Figure 20: Distribution of magnetic field and arrows indicating the direction of the electric field for the 0.3-THz incident wave. (a) and (c) are results from the optimal configuration shown in Fig.19 (a). (b) and (d) are results from the optimal configuration shown in Fig

45 15 10 Target frequency Re(µ eff ) of Fig.15(a) -10 Im(µ ) of Fig.15(a) eff -15 Re(µ ) of Fig.15(b) eff Im(µ ) of Fig.15(b) eff -20 2x x x x10 11 Frequency Figure 21: Effective permeability of the optimal configurations with an incident wave from 0.2 to 0.5 THz Design of dielectric ring resonator As our final numerical example, we optimize a dielectric ring resonator. To perform iterating numerical optimization based on the finite element method, we need the target WG eigenmode to be automatically selected from the numerous resulting eigenmodes. We introduce the two-step analysis proposed in [72] to specify the target mode during the iteration using the analysis domains shown in Fig.22. First, we solve the small closed finiteelement model composed of a device surrounded by a perfect magnetic wall, as drawn in Fig.22(a), to obtain the eigenfrequency of the target WG mode. The first eigenmode of the model corresponds to the target WG mode. Second, we solve for the original model surrounded by PML domains, as drawn in Fig.22(b), specifying the target eigenfrequency obtained by the closed model. In the design domain, the relative permittivity is formulated the same way as in the third example using Eq.(59). Optimizations are performed with specified TM modes (p = 1, M = 11). The design domain is meshed 45

46 with square elements. The other domains are meshed with triangular elements. The diffusion coefficient κ is set to 0.1 l 2. Assuming horizontal mirror symmetry of the optimal shapes, we only optimize the upper half of the design domain. Thus, the 3750 design variables are updated during optimization. The medium constituting the resonator is assumed to be GaAlAs with ϵ 1 = We maximize the Q factor and minimize the mode volume V using Eqs.(31) and (32), respectively. The phase field function is updated by using the SLP based method of Eqs.(55)-(57) to handle the equality constraint directory. The initial shape for both optimizations is shown in Fig.23. We chose it for the ease of mode volume optimization since its initial dependency is stronger than that of the Q factor optimization because Eq.(32) handles the local maximum energy. 46

47 z Center axis z Center axis Air domain Air domain 2µm 3µm 2µm 6µm Design domain r Design domain r 1.5µm 1.5µm 1.8µm 4µm Perfect magnetic wall PML Domain (0.25µm) (a) (b) Figure 22: Analysis domain for designing the dielectric ring resonator including the design domain. (a) Closed model only used for finding the frequency corresponding to the target WG mode. (b) Open model used for the optimization. 47

48 Figure 23: Initial shapes for designing the dielectric ring resonator. Figures 25 and 26 show the optimal configuration, the distribution of the electric energy density ϵ E 2 and the electric field intensity E 2 for each optimization. These figures are shown in the design domain, which is a 1.5 µm 2 µm box whose left side corresponds to the center axis. The procedure needed 100 and 30 iterations until convergence. The resulting Q factor, mode volume and wavelength are Q rad = , V mode = , λ = 1207nm and Q rad = , V mode = , λ = 1200nm in Figs.25 and 26 respectively. These results are similar to those obtained in [67], although the two-step optimization composed of topology optimization and phase field in [67] differs from the proposed framework. In figure 25, the optimal shape has a large smooth convex form covering the electric-field hot spot to reduce radiation losses. In contrast to high-q-factor shape, the lowmode-volume optimal shape shown in Fig.26 has a small concave form near the center to enhance the maximum electric energy. The variations of the Q factor and constraint after extracting black and white shapes are 4.49% and 0.00%, respectively, in Fig.25. The variation of the mode volume and 48

49 constraint after extracting black and white shapes are 12.15% and 1.67%, respectively, in Fig.26. Since the center of the optimal configuration is very thin, the density of one element has a large effect on the performance of the shape with optimal mode volume. 7x x10-6 2x x10-6 Q factor 6x10 7 5x10 7 4x10 7 3x10 7 Target wavelength 1.2x10-6 9x10-7 6x10-7 2x10 7 3x10-7 1x10 7 Q factor Wavelength Iteration Wavelength Mode volume 1.5x x10-19 Target wavelength 1.2x10-6 9x10-7 6x10-7 5x x10-7 Mode volume Wavelength Iteration Wavelength (a) (b) Figure 24: Convergence history of (a) Q factor and (b) mode volume of the dielectric ring resonator optimization. 49

50 (a) (b) (c) Figure 25: Optimal shapes obtained by maximizing Q rad. (a) Optimal shape. (b) Electric field intensity E 2 distribution. (c) Electric energy density ϵ E 2 distribution. (a) (b) (c) Figure 26: Optimal shapes obtained by maximizing V mode. (a) Optimal shape. (b) Electric field intensity E 2 distribution. (c) Electric energy density ϵ E 2 distribution. 50